Asymptotic Upper and Lower Estimates of a Class of Positive Solutions of a Discrete Linear Equation with a Single Delay

Abstract

We study a frequently investigated class of linear difference equationsΔv(n)=−p(n)v(n−k)with a positive coefficientp(n)and a single delayk. Recently, it was proved that if the functionp(n)is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the functionp(n)is bounded above and below by certain functions.

Existence and Convergence of the Positive Solutions of a Discrete Epidemic Model

“…Over the past decades, there has been an increasing interest in studying nonlinear and rational difference equations arising in applied sciences, and hence there are many papers and books concerning theory and applications of difference equations; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein. The existence of positive solutions for difference equations has also received a great deal of attention in the literature.…”

“…The existence of positive solutions for difference equations has also received a great deal of attention in the literature. To identify a few, we refer the reader to Baštinec et al [1,2], Diblík and Hlavičková [6], and the references therein. In [11], the authors posed some research projects about the behavior of solutions of the following deterministic epidemic models:…”